4.2 Standard situations to evaluate the quasi-local quantities

There are exact solutions to the Einstein equations and classes of special (e.g., asymptotically flat)
spacetimes in which there is a commonly accepted definition of energy-momentum (or at least mass) and
angular momentum. In this section we review these situations and recall the definition of these ‘standard’
expressions.

4.2.1 Round spheres

If the spacetime is spherically symmetric, then a two-sphere, which is a transitivity surface of the
rotation group, is called a round sphere. Then in a spherical coordinate system
the spacetime metric takes the form , where
and are functions of and . (Hence, is called the area-coordinate.) Then,
with the notation of Section 4.1, one obtains . Based on the
investigations of Misner, Sharp, and Hernandez [365, 267], Cahill and McVitte [122] found

to be an appropriate (and hence, suggested to be the general) notion of energy, the Misner–Sharpenergy, contained in the two-sphere . (For another expression of
in terms of the norm of the Killing fields and the metric, see [577].) In particular,
for the Reissner–Nordström solution , while for the isentropic fluid
solutions , where and are the usual parameters of the
Reissner–Nordström solutions and is the energy density of the fluid [365, 267] (for the static solution,
see, e.g., Appendix B of [240]). Using Einstein’s equations, simple equations can be derived
for the derivatives and , and if the energy-momentum tensor satisfies
the dominant energy condition, then . Thus, is a monotonic function
of , providedis the area-coordinate. Since, by spherical symmetry all the quantities
with nonzero spin weight, in particular the shears and , are vanishing and is
real, by the GHP form of Eqs. (4.3), (4.4) the energy function can also be written as

Any of these expressions is considered to be the ‘standard’ definition of the energy for round
spheres.4
The last of these expressions does not depend on whether is an area-coordinate or not.

contains a contribution from the gravitational ‘field’ too. For example, for fluids it is
not simply the volume integral of the energy density of the fluid, because that would be
. This deviation can be interpreted as the contribution of the gravitational potential
energy to the total energy. Consequently, is not a globally monotonic function of ,
even if . For example, in the closed Friedmann–Robertson–Walker spacetime (where,
to cover the whole three-space, cannot be chosen to be the area–radius and )
is increasing for , taking its maximal value at , and decreasing for
.

This example suggests a slightly more exotic spherically-symmetric spacetime. Its spacelike slice will
be assumed to be extrinsically flat, and its intrinsic geometry is the matching of two conformally flat
metrics. The first is a ‘large’ spherically-symmetric part of a hypersurface of the closed
Friedmann–Robertson–Walker spacetime with the line element , where is the line
element for the flat three-space and with positive constants and , and the
range of the Euclidean radial coordinate is , where . It contains a maximal
two-surface at with round-sphere mass parameter . The scalar
curvature is , and hence, by the constraint parts of the Einstein equations and by the
vanishing of the extrinsic curvature, the dominant energy condition is satisfied. The other metric is the
metric of a piece of a hypersurface in the Schwarzschild solution with mass parameter
(see [213]): , where and the Euclidean radial coordinate runs from
to , where . In this geometry there is a minimal surface at , the scalar
curvature is zero, and the round-sphere energy is . These two metrics can be matched to
obtain a differentiable metric with a Lipschitz-continuous derivative at the two-surface of the matching
(where the scalar curvature has a jump), with arbitrarily large ‘internal mass’ and arbitrarily
small ADM mass . (Obviously, the two metrics can be joined smoothly, as well, by an
‘intermediate’ domain between them.) Since this space looks like a big spherical bubble on a
nearly flat three-plane – like the capital Greek letter – for later reference we will call it
an‘-spacetime’.

Spherically-symmetric spacetimes admit a special vector field, called the Kodama vector field , such
that is divergence free [321]. In asymptotically flat spacetimes is timelike in the
asymptotic region, in stationary spacetimes it reduces to the Killing symmetry of stationarity (in
fact, this is hypersurface-orthogonal), but, in general, it is not a Killing vector. However, by
, the vector field has a conserved flux on a spacelike hypersurface
. In particular, in the coordinate system and in the line element given in the
first paragraph above . If is a solid ball of radius , then
the flux of is precisely the standard round-sphere expression (4.7) for the two-sphere
[375].

An interesting characterization of the dynamics of the spherically-symmetric gravitational fields can be
given in terms of the energy function given by (4.7) (or by (4.8)) (see, e.g., [578, 352, 250]). In
particular, criteria for the existence and formation of trapped surfaces and for the presence and nature of
the central singularity can be given by . Other interesting quasi-locally–defined quantities
are introduced and used to study nonlinear perturbations and backreaction in a wide class of
spherically-symmetric spacetimes in [483]. For other applications of in cosmology see, e.g.,
[484, 130].

4.2.2 Small surfaces

In the literature there are two kinds of small surfaces. The first is that of the small spheres (both in the
light cone of a point and in a spacelike hypersurface), introduced first by Horowitz and Schmidt [275],
and the other is the concept of small ellipsoids in a spacelike hypersurface, considered first
by Woodhouse in [313]. A small sphere in the light cone is a cut of the future null cone in
the spacetime by a spacelike hypersurface, and the geometry of the sphere is characterized by
data at the vertex of the cone. The sphere in a hypersurface consists of those points of a given
spacelike hypersurface, whose geodesic distance in the hypersurface from a given point ,
the center, is a small given value, and the geometry of this sphere is characterized by data at
this center. Small ellipsoids are two-surfaces in a spacelike hypersurface with a more general
shape.

To define the first, let be a point, and a future-directed unit timelike vector at . Let
, the ‘future null cone of in ’ (i.e., the boundary of the chronological future of ).
Let be the future pointing null tangent to the null geodesic generators of , such that, at the vertex
, . With this condition we fix the scale of the affine parameter on the different generators,
and hence, by requiring , we fix the parametrization completely. Then, in an open neighborhood
of the vertex , is a smooth null hypersurface, and hence, for sufficiently small , the set
is a smooth spacelike two-surface and is homeomorphic to . is called a
small sphere of radius with vertex . Note that the condition fixes the boost gauge,
too.

Completing to get a Newman–Penrose complex null tetrad such that the
complex null vectors and are tangent to the two-surfaces , the components of
the metric and the spin coefficients with respect to this basis can be expanded as a series in
. If, in addition, the spinor constituent of is required to be parallelly
propagated along , then the tetrad becomes completely fixed, yielding the vanishing of
several (combinations of the) spin coefficients. Then the GHP equations can be solved with any
prescribed accuracy for the expansion coefficients of the metric on , the GHP spin
coefficients , , , , and , and the higher-order expansion coefficients of
the curvature in terms of the lower-order curvature components at . Hence, the expression
of any quasi-local quantity for the small sphere can be expressed as a series of
,

where the expansion coefficients are still functions of the coordinates, or ,
on the unit sphere . If the quasi-local quantity is spacetime-covariant, then the unit
sphere integrals of the expansion coefficients must be spacetime covariant expressions
of the metric and its derivatives up to some finite order at and the ‘time axis’ . The
necessary degree of the accuracy of the solution of the GHP equations depends on the
nature of and on whether the spacetime is Ricci-flat in the neighborhood of or
not.5
These solutions of the GHP equations, with increasing accuracy, are given in [275, 313, 118, 494].

Obviously, we can calculate the small-sphere limit of various quasi-local quantities built from the matter
fields in the Minkowski spacetime, as well. In particular [494], the small-sphere expressions for the
quasi-local energy-momentum and the (anti-self-dual part of the) quasi-local angular momentum of the
matter fields based on , are, respectively,

where , , is the ‘Cartesian spin frame’ at and the origin of the Cartesian coordinate
system is chosen to be the vertex . Here can be interpreted as the translation
one-forms, while is an average on the unit sphere of the boost-rotation Killing
one-forms that vanish at the vertex . Thus, and are the three-volume times the
energy-momentum and angular momentum density with respect to , respectively, that the observer with
four-velocity sees at .

Interestingly enough, a simple dimensional analysis already shows the structure of the leading terms in
a large class of quasi-local spacetime covariant energy-momentum and angular momentum
expressions. In fact, if is any coordinate-independent quasi-local quantity built from the first
derivatives of the spacetime metric, then in its expansion the difference of the power of
and the number of the derivatives in every term must be one, i.e., it must have the form

where , , are scalars. They are polynomial expressions of , and
at the vertex , and they depend linearly on the tensors that are constructed at from ,
and linearly from the coordinate-dependent quantities , , …. Since there is no nontrivial tensor built
from the first derivative and , the leading term is of order . Its coefficient
must be a linear expression of and , and polynomial in , and . In particular, if
is to represent energy-momentum with generator at , then the leading term must be

for some unspecified constants , , and , where , the projection to
the subspace orthogonal to . If, in addition to the coordinate-independence of , it is
Lorentz-covariant, i.e., it does not, for example, depend on the choice for a normal to (e.g., in the
small-sphere approximation on ) intrinsically, then the different terms in the above expression
must depend on the boost gauge of the external observer in the same way. Therefore,
, in which case the first and the third terms can in fact be written as .
Then, comparing Eq. (4.11) with Eq. (4.9), we see that , and hence the term
would have to be interpreted as the contribution of the gravitational ‘field’ to the
quasi-local energy-momentum of the matter + gravity system. However, this contributes only
to energy, but not to linear momentum in any frame defined by the observer , even in a
general spacetime. This seems to be quite unacceptable. Furthermore, even if the matter fields
satisfy the dominant energy condition, given by Eq. (4.11) can be negative, even for
, unless . Thus, in the leadingorder in nonvacuum, any coordinate andLorentz-covariant quasi-local energy-momentum expression which is nonspacelike and futurepointing, should be proportional to the energy-momentum density of the matter fields seen by theobservertimes the Euclidean volume of the three-ball of radius. No contribution from the
gravitational ‘field’ is expected at this order. In fact, this result is compatible the with the principle of
equivalence, and the particular results obtained in the relativistically corrected Newtonian theory
(considered in Section 3.1.1) and in the weak field approximation (see Sections 4.2.5 and 7.1.1
below). Interestingly enough, even for a timelike Killing field , the well known expression of
Komar does not satisfy this criterion. (For further discussion of Komar’s expression see also
Section 12.1.)

If the neighborhood of is vacuum, then the -order term is vanishing, and the fourth-order
term must be built from . However, the only scalar polynomial expression of ,
, , and the generator vector , depending linearly on the latter two,
is the zero tensor field. Thus, the -order term in vacuum is also vanishing. At the fifth
order the only nonzero terms are quadratic in the various parts of the Weyl tensor, yielding

for constants , , , and , where is the electric part and
is the magnetic part of the Weyl curvature, and is
the induced volume 3-form. However, using the identities ,
, and , we can
rewrite the above formula to be

Again, if does not depend on intrinsically, then , in which case the first and the
fourth terms together can be written into the Lorentz covariant form . In a general
expression the curvature invariants and may be present. Since, however,
and at a given point are independent, these invariants can be arbitrarily large positive or negative,
and hence, for or the quasi-local energy-momentum could not be future pointing and
nonspacelike. Therefore, in vacuum in the leadingorder any coordinate and Lorentz-covariantquasi-local energy-momentum expression, which is nonspacelike and future pointing must be proportional tothe Bel–Robinson ‘momentum’.

Obviously, the same analysis can be repeated for any other quasi-local quantity. For the energy-momentum,
has the structure , for angular momentum it is , while the area of
is . Therefore, the leading term in the expansion of the angular momentum is and
order in nonvacuum and vacuum with the energy-momentum and the Bel–Robinson tensors, respectively,
while the first nontrivial correction to the area is of order and in nonvacuum and vacuum,
respectively.

On the small geodesic sphere of radius in the given spacelike hypersurface one can
introduce the complex null tangents and above, and if is the future-pointing unit normal of
and the outward directed unit normal of in , then we can define and
. Then is a Newman–Penrose complex null tetrad, and the relevant
GHP equations can be solved for the spin coefficients in terms of the curvature components at
.

The small ellipsoids are defined as follows [313]. If is any smooth function on with a
nondegenerate minimum at with minimum value , then, at least on an open
neighborhood of in , the level surfaces are smooth
compact two-surfaces homeomorphic to . Then, in the limit, the surfaces look
like small nested ellipsoids centered at . The function is usually ‘normalized’ so that
.

A slightly different framework for calculations in small regions was used in [327, 170, 235]. Instead of
the Newman–Penrose (or the GHP) formalism and the spin coefficient equations, holonomic (Riemann or
Fermi type normal) coordinates on an open neighborhood of a point or a timelike curve
are used, in which the metric, as well as the Christoffel symbols on , are expressed by the coordinates on
and the components of the Riemann tensor at or on . In these coordinates and the
corresponding frames, the various pseudotensorial and tetrad expressions for the energy-momentum
have been investigated. It has been shown that a quadratic expression of these coordinates
with the Bel–Robinson tensor as their coefficient appears naturally in the local conservation
law for the matter energy-momentum tensor [327]; the Bel–Robinson tensor can be recovered
as some ‘double gradient’ of a special combination of the Einstein and the Landau–Lifshitz
pseudotensors [170]; Møller’s tetrad expression, as well as certain combinations of several other classical
pseudotensors, yield the Bel–Robinson tensor [473, 470, 471]. In the presence of some non-dynamical
(background) metric a 11-parameter family of combinations of the classical pseudotensors exists,
which, in vacuum, yields the Bel–Robinson tensor [472, 474]. (For this kind of investigation see
also [465, 468, 466, 467, 469]).

In [235] a new kind of approximate symmetries, namely approximate affine collineations, are introduced
both near a point and a world line, and used to introduce Komar-type ‘conserved’ currents. (For a readable
text on the non-Killing type symmetries see, e.g., [233].) These symmetries turn out to yield a
nontrivial gravitational contribution to the matter energy-momentum, even in the leading
order.

4.2.3 Large spheres near spatial infinity

Near spatial infinity we have the a priori and falloff for the three-metric and
extrinsic curvature , respectively, and both the evolution equations of general relativity
and the conservation equation for the matter fields preserve these conditions. The
spheres of coordinate radius in are called large spheres if the values of are
large enough, such that the asymptotic expansions of the metric and extrinsic curvature are
legitimate.6
Introducing some coordinate system, e.g., the complex stereographic coordinates, on one sphere and then
extending that to the whole along the normals of the spheres, we obtain a coordinate system
on . Let , , be a GHP spinor dyad on adapted to the large
spheres in such a way that and are tangent to the spheres and
, the future directed unit normal of . These conditions fix the spinor dyad
completely, and, in particular, , the outward directed unit normal to the spheres
tangent to .

The falloff conditions yield that the spin coefficients tend to their flat spacetime value as and the
curvature components to zero like . Expanding the spin coefficients and curvature components as a
power series of , one can solve the field equations asymptotically (see [65, 61] for a different
formalism). However, in most calculations of the large sphere limit of the quasi-local quantities,
only the leading terms of the spin coefficients and curvature components appear. Thus, it is
not necessary to solve the field equations for their second or higher-order nontrivial expansion
coefficients.

Using the flat background metric and the corresponding derivative operator we can
define a spinor field to be constant if . Obviously, the constant spinors form
a two–complex-dimensional vector space. Then, by the falloff properties .
Thus, we can define the asymptotically constant spinor fields to be those that satisfy
, where is the intrinsic Levi-Civita derivative operator on . Note
that this implies that, with the notation of Eq. (4.6), all the chiral irreducible parts, ,
, , and of the derivative of the asymptotically constant spinor field are
.

4.2.4 Large spheres near null infinity

Let the spacetime be asymptotically flat at future null infinity in the sense of Penrose [413, 414, 415, 426]
(see also [208]), i.e., the physical spacetime can be conformally compactified by an appropriate boundary
. Then future null infinity will be a null hypersurface in the conformally rescaled
spacetime. Topologically it is , and the conformal factor can always be chosen such that the
induced metric on the compact spacelike slices of is the metric of the unit sphere. Fixing
such a slice (called ‘the origin cut of ’) the points of can be labeled by a
null coordinate, namely the affine parameter along the null geodesic generators of
measured from and, for example, the familiar complex stereographic coordinates
, defined first on the origin cut and then extended in a natural way along the null
generators to the whole . Then any other cut of can be specified by a function
. In particular, the cuts are obtained from by a pure time
translation.

The coordinates can be extended to an open neighborhood of in the spacetime in the
following way. Let be the family of smooth outgoing null hypersurfaces in a neighborhood of ,
such that they intersect the null infinity just in the cuts , i.e., . Then let be the
affine parameter in the physical metric along the null geodesic generators of . Then forms
a coordinate system. The , two-surfaces (or simply if no confusion can
arise) are spacelike topological two-spheres, which are called large spheres of radius near future null
infinity. Obviously, the affine parameter is not unique, its origin can be changed freely:
is an equally good affine parameter for any smooth . Imposing certain
additional conditions to rule out such coordinate ambiguities we arrive at a ‘Bondi-type coordinate
system’.7
In many of the large-sphere calculations of the quasi-local quantities the large spheres should be assumed to
be large spheres not only in a general null, but in a Bondi-type coordinate system. For a detailed discussion
of the coordinate freedom left at the various stages in the introduction of these coordinate systems, see, for
example, [394, 393, 107].

In addition to the coordinate system, we need a Newman–Penrose null tetrad, or rather a GHP spinor
dyad, , , on the hypersurfaces . (Thus, boldface indices are referring to the
GHP spin frame.) It is natural to choose such that be the tangent
of the null geodesic generators of , and itself be constant along . Newman and
Unti [394] chose to be parallelly propagated along . This choice yields the vanishing of a
number of spin coefficients (see, for example, the review [393]). The asymptotic solution of
the Einstein–Maxwell equations as a series of in this coordinate and tetrad system is
given in [394, 179, 425], where all the nonvanishing spin coefficients and metric and curvature
components are listed. In this formalism the gravitational waves are represented by the -derivative
of the asymptotic shear of the null geodesic generators of the outgoing null hypersurfaces
.

From the point of view of the large sphere calculations of the quasi-local quantities, the choice of
Newman and Unti for the spinor basis is not very convenient. It is more natural to adapt the GHP spin
frame to the family of the large spheres of constant ‘radius’ , i.e., to require and
to be tangents of the spheres. This can be achieved by an appropriate null rotation
of the Newman–Unti basis about the spinor . This rotation yields a change of the spin
coefficients and the metric and curvature components. As far as the present author is aware, the
rotation with the highest accuracy was done for the solutions of the Einstein–Maxwell system by
Shaw [455].

In contrast to the spatial-infinity case, the ‘natural’ definition of the asymptotically constant spinor
fields yields identically zero spinors in general [106]. Nontrivial constant spinors in this sense could exist
only in the absence of the outgoing gravitational radiation, i.e., when . In the language of
Section 4.1.7, this definition would be , ,
and . However, as Bramson showed [106], half of these conditions can be
imposed. Namely, at future null infinity (and at past null infinity
) can always be imposed asymptotically, and has two linearly-independent
solutions , , on (or on , respectively). The space of its solutions turns out
to have a natural symplectic metric , and we refer to as future asymptotic spin space. Its
elements are called asymptotic spinors, and the equations , the future/past
asymptotic twistor equations. At asymptotic spinors are the spinor constituents of the
BMS translations: Any such translation is of the form for
some constant Hermitian matrix . Similarly, (apart from the proper supertranslation
content) the components of the anti-self-dual part of the boost-rotation BMS vector fields are
, where are the standard Pauli matrices (divided by ) [496].
Asymptotic spinors can be recovered as the elements of the kernel of several other operators
built from , , , and , too. In the present review we use only the fact that
asymptotic spinors can be introduced as antiholomorphic spinors (see also Section 8.2.1), i.e.,
the solutions of (and at past null infinity as holomorphic spinors),
and as special solutions of the two-surface twistor equation (see also
Section 7.2.1). These operators, together with others reproducing the asymptotic spinors, are discussed
in [496].

The Bondi–Sachs energy-momentum given in the Newman–Penrose formalism has already
become its ‘standard’ form. It is the unit sphere integral on the cut of a combination of
the leading term of the Weyl spinor component , the asymptotic shear and its
-derivative, weighted by the first four spherical harmonics (see, for example, [393, 426]):

where , , are the -component of the vectors of a spin frame in the space
of the asymptotic spinors. (For the various realizations of these spinors see, e.g., [496].) The
minimal assumptions on the physical Ricci tensor that already ensure that the Bondi–Sachs
energy-momentum and Bondi’s mass-loss are well defined are determined by Tafel [505]. The
expression of the Bondi–Sachs energy-momentum in terms of the conformal factor is also given
there.

Similarly, the various definitions for angular momentum at null infinity could be rewritten in this
formalism. Although there is no generally accepted definition for angular momentum at null infinity in
general spacetimes, in stationary and in axi-symmetric spacetimes there is. The former is the unit sphere
integral on the cut of the leading term of the Weyl spinor component , weighted by appropriate
(spin-weighted) spherical harmonics:

In particular, Bramson’s expression also reduces to this ‘standard’ expression in the absence of the outgoing
gravitational radiation [109]. If the spacetime is axi-symmetric, then the generally accepted definition of
angular momentum is that of Komar with the numerical coefficient (rather than ) and
in (3.15). This view is supported by the partial results of a quasi-local canonical analysis of general
relativity given in [499], too.

Instead of the Bondi type coordinates above, one can introduce other ‘natural’ coordinates in a
neighborhood of . Such is the one based on the outgoing asymptotically–shear-free null geodesics [27].
While the Bondi-type coordinate system is based on the null geodesic generators of the outgoing null
hypersurfaces , and hence, in the rescaled metric these generators are orthogonal to the cuts , the
new coordinate system is based on the use of outgoing null geodesic congruences that extend to
but are not orthogonal to the cuts of (and hence, in general, they have twist). The
definition of the new coordinates is analogous to that of the Bondi-type coordinates:
labels the intersection point of the actual geodesic and , while is the affine
parameter along the geodesic. The tangent of this null congruence is asymptotically null rotated
about : In the NP basis above , where
and is a complex valued function (with spin
weight one) on . Then Aronson and Newman show in [27] that if is chosen to satisfy
, then the asymptotic shear of the congruence is, in fact, of order , and by an
appropriate choice for the other vectors of the NP basis many spin coefficients can be made
zero. In this framework it is the function that plays a role analogous to that of , and,
indeed, the asymptotic solution of the field equations is given in terms of in [27]. This
can be derived from the solution of the good-cut equation, which, however, is not
uniquely determined, but depends on four complex parameters: . It is this
freedom that is used in [325, 326] to introduce the angular momentum at future null infinity (see
Section 3.2.4). Further discussion of these structures, in particular their connection with the
solutions of the good-cut equation and the -space, as well as their applications, is given
in [324, 325, 326, 5].

4.2.5 Other special situations

In the weak field approximation of general relativity [525, 36, 534, 426, 303] the gravitational field is
described by a symmetric tensor field on Minkowski spacetime , and the dynamics of the
field is governed by the linearized Einstein equations, i.e., essentially the wave equation.
Therefore, the tools and techniques of the Poincaré-invariant field theories, in particular the
Noether–Belinfante–Rosenfeld procedure outlined in Section 2.1 and the ten Killing vectors of the
background Minkowski spacetime, can be used to construct the conserved quantities. It turns out that
the symmetric energy-momentum tensor of the field is essentially the second-order term
in the Einstein tensor of the metric . Thus, in the linear approximation the
field does not contribute to the global energy-momentum and angular momentum of the
matter + gravity system, and hence these quantities have the form (2.5) with the linearized
energy-momentum tensor of the matter fields. However, as we will see in Section 7.1.1, this
energy-momentum and angular momentum can be re-expressed as a charge integral of the (linearized)
curvature [481, 277, 426].

pp-waves spacetimes are defined to be those that admit a constant null vector field , and they
interpreted as describing pure plane-fronted gravitational waves with parallel rays. If matter is present, then
it is necessarily pure radiation with wave-vector , i.e., holds [478]. A remarkable feature of
the pp-wave metrics is that, in the usual coordinate system, the Einstein equations become a
two-dimensional linear equation for a single function. In contrast to the approach adopted almost
exclusively, Aichelburg [8] considered this field equation as an equation for a boundary value problem. As we
will see, from the point of view of the quasi-local observables this is a particularly useful and natural
standpoint. If a pp-wave spacetime admits an additional spacelike Killing vector with closed
orbits, i.e., it is cyclically symmetric too, then and are necessarily commuting and are
orthogonal to each other, because otherwise an additional timelike Killing vector would also be
admitted [485].

Since the final state of stellar evolution (the neutron star or black hole state) is expected to be described
by an asymptotically flat, stationary, axisymmetric spacetime, the significance of these spacetimes is
obvious. It is conjectured that this final state is described by the Kerr–Newman (either outer or black hole)
solution with some well-defined mass, angular momentum and electric charge parameters [534]. Thus,
axisymmetric two-surfaces in these solutions may provide domains, which are general enough but for which
the quasi-local quantities are still computable. According to a conjecture by Penrose [418],
the (square root of the) area of the event horizon provides a lower bound for the total ADM
energy. For the Kerr–Newman black hole this area is . Thus,
particularly interesting two-surfaces in these spacetimes are the spacelike cross sections of the event
horizon [80].

There is a well-defined notion of total energy-momentum not only in the asymptotically flat, but even in
the asymptotically anti-de Sitter spacetimes as well. This is the Abbott–Deser energy [1], whose
positivity has also been proven under similar conditions that we had to impose in the positivity
proof of the ADM energy [220]. (In the presence of matter fields, e.g., a self-interacting scalar
field, the falloff properties of the metric can be weakened such that the ‘charges’ defined at
infinity and corresponding to the asymptotic symmetry generators remain finite [265].) The
conformal technique, initiated by Penrose, is used to give a precise definition of the asymptotically
anti-de Sitter spacetimes and to study their general, basic properties in [42]. A comparison and
analysis of the various definitions of mass for asymptotically anti-de Sitter metrics is given
in [150].

Extending the spinorial proof [349] of the positivity of the total energy in asymptotically anti-de Sitter
spacetime, Chruściel, Maerten and Tod [149] give an upper bound for the angular momentum and
center-of-mass in terms of the total mass and the cosmological constant. (Analogous investigations show
that there is a similar bound at the future null infinity of asymptotically flat spacetimes with no outgoing
energy flux, provided the spacetime contains a constant–mean-curvature, hyperboloidal, initial-data set on
which the dominant energy condition is satisfied. In this bound the role of the cosmological constant is
played by the (constant) mean curvature of the hyperboloidal spacelike hypersurface [151].)
Thus, it is natural to ask whether or not a specific quasi-local energy-momentum or angular
momentum expression has the correct limit for large spheres in asymptotically anti-de Sitter
spacetimes.

"Quasi-Local Energy-Momentum and Angular Momentum in
General Relativity"